Depending on the setting, this formulation can be made rigorous in various ways: for example via an analytical approach on configuration spaces as done in [1]}, which in fact models infinite configurations of particles over \(\mathbb {R}^d\) , or via martingale techniques with \(\mathcal {T}\)  a closed subset of \(\mathbb {R}^d\)  and \(\gamma ={L}^d|_{\mathcal {T}}\)  (see [2]}). Moreover, in the latter, under the assumption of continuous, bounded, and integrable mutation/competition kernels, it is also shown that the process converges in the large-population limit \(n\rightarrow \infty \)  to the mean-field equation (REF ), which can be rewritten as
\(\partial _t \nu _t =\kappa ^+[\nu _t]-\kappa ^-[\nu _t], \qquad \nu _t\in \Gamma .\qquad \mathrm {(\sf MF)}\) 
